X-ray tensor tomography (XTT) is an imaging modality for reconstructing three-dimensional (3D) scattering tensors from a sequence of two-dimensional (2D) dark-field projections obtained in a grating interferometer. Due to the use of anisotropic dark-field signal XTT allows to provide information about the orientations of micro-scale structures which cannot be resolved in traditional X-ray imaging techniques, such as absorption contrast X-ray tomography and differential phase contrast (DPC) X-ray tomography.
XTT allows for non-destructive testing and can be used, for example, to resolve the orientations of fibers in dense carbon fiber composites. These materials are critical to the safety, weight and performance of automobiles and airplanes. Another exemplary application of XTT is to resolve the structure in polymer materials manufactured using injection molding. In addition, XTT can be used to resolve the structures of dentinal tubules in human tooth which are not directly visible in traditional X-ray imaging and to resolve the orientations of the structures within bones which allows to study the strength mechanisms of bones.
An Example of a state of the art XTT-system 10 including an X-ray grating interferometer in a so called Talbot-Lau configuration is shown in FIG. 1 and provided in the Article “X-ray tensor tomography”, A. Malecki et at, EPL (Europhysics Letters), vol. 105, no. 3, p. 38002, 2011. The XTT-system 10 comprises an X-ray source 12, a source grating 14, a first grating 16, a second grating 18, a specimen stage 20 and a detector 22.
Such a system 10 can provide three types of projection images of the specimen: absorption contrast, differential phase contrast and dark-field contrast. All three contrast modes provide complimentary information about the inside structure of the specimen. The absorption contrast can provide information about the 3D distribution of the absorption coefficient, which may be related to the imaginary part of the refractive index, for example, and the DPC can provide information about the 3D distribution of the real part of the refractive index within a specimen. Both in X-ray absorption contrast computer tomography (CT) and in DPC CT scalar quantities, which are respectively related to the absorption coefficient and to the refractive index, are reconstructed by using reconstruction algorithms for a plurality of volume elements or voxels of the specimen.
For the case of XTT a plurality of tensor quantities are reconstructed for a plurality of associated voxels, wherein each tensor quantity corresponds to a scattering tensor providing information about the directional distribution of the scattering power at the respective voxel. The scattering tensors are reconstructed from a sequence of 2D X-ray dark-field projections. The contrast in the dark-field projection images is caused by ultra-small angle scattering of X-rays by small structures, which can have a size in the sub-micron and micron range, and thus encode information about the 3D position and the 3D orientation of the structures within the specimen. The reconstruction of full 3D scattering tensors for a specimen is known in the art and describes, for example, in the article “Constrained X-ray tensor tomography reconstruction”, J. Vogel et al., Opt. Express, vol. 23, no 12, pp. 15134-15151, 2015.
For the purpose of XTT reconstruction several dark-field projections are acquired for different orientations of the specimen with respect to the XTT-system 10, wherein for each orientation the first and second gratings 16, 18 are shifted with respect to each other into different shifting positions, for each of which a projection is acquired with the detector 20. The projections corresponding to the different shifting positions of the gratings are combined to calculate absorption, differential phase contrast and dark-field signal for each sample orientation. The orientation of the specimen (not shown) is controlled by using the specimen stage 20. In the XTT-system 10 the specimen stage 20 corresponds to a three-circle Eulerian Cradle which allows rotating the specimen                about a first axis of rotation 24 into different angle positions ϕ,        about a second axis of rotation 26 into different angle positions θ and        about a third axis of rotation 28 into different angle positions ψ.        
For the acquisitions the sample is positioned at a center of the Eulerian Cradle on an optical axis 30 of the MT-system 10, wherein the center is intersected by all three axes 24, 26, 28.
As shown in FIG. 1, the direction of the first axis of rotation 24 coincides with the y′-direction of a coordinate system x′,y′,z′ of the specimen. The x′,y′,z′ coordinate system of the specimen is rotated together with the specimen with respect to a stationary coordinate system x,y,z of the XTT-system 10. The first axis of rotation 24 lies within a plane defined by the Eulerian Cradle 20 and is rotated in this plane during a rotation about the second axis of rotation 26. The direction of the second axis of rotation 26 is orthogonal to the plane defined by the Eulerian Cradle, such that the second axis of rotation 26 is always orthogonal to the first axis of rotation 24. For ϕ=0° or 180° the direction of the second axis of rotation 26 corresponds to the z′-direction.
The direction of the third axis of rotation 28 coincides with the y-direction, such that the third axis of rotation 28 is stationary with respect to the XTT-system 10 and always orthogonal to the second axis of rotation 26.
In case of ψ=0°, the second axis of rotation 26 coincides with the optical axis 30 of the XTT-system 10 (z-direction). In case of θ=0°, the direction of the first axis of rotation 24 (y′-direction) coincides with the y-direction.
All gratings 14, 16, 18 are pairwise parallel and therefore orthogonal to the optical axis 30. Each of the gratings 14, 16, 18 has grating lines, wherein the grating lines of each of the gratings 14, 16, 18 extend in the same direction orthogonally to the optical axis 30. A sensitivity direction 32 of the XTT-system to, which corresponds to the direction in which a phase shift of the X-rays is measured, is parallel to the planes of the gratings 14, 16, 18 and orthogonal to the grating lines. In the XTT-system 10 of FIG. 1, the sensitivity direction 32 coincides with the y-direction.
The acquisition scheme M proposed in the above mentioned article by Malecki et al. is given by:M={(ψ,θ,ϕ);ψ∈[0°,45°,90°,135°],θ∈[0°,45°,90°,135°],ϕ∈[−36,67°,−36°, . . . ,36,67°]}resulting in 1776 different acquisition poses or orientations of the specimen, wherein for each orientation the first grating 16 is scanned or shifted with respect to the second grating 18 in eight steps over one grating period in the sensitivity direction, such that 14208 projections are acquired in total.
This acquisition with the above described acquisition geometry of the XTT-system 10 of FIG. 1 allows illuminating the specimen in a variety of different orientations, such that for the volume of the specimen a set of 3D scattering tensors can be reconstructed, wherein each scattering tensor provides information about an orientation of a scattering structure within the specimen within 3D space at a corresponding position in 3D space.
However, the space requirements of the acquisition geometry of the XTT-system 10 of FIG. 1 are quite high and the acquisition time is quite long, which leads to challenges to the commercialization of XTT.